Research On Iteration Method Of Two Specific Complex Linear Systems

Many practical problems are ultimately attributed to solving complex linear systems in many fields of scientific engineering calculation.Normally,the iteration method is used to solve these problems,but it still contributes to more iteration time and even non-convergence.The precondition-ed technique,parameter acceleration technique and SOR acceleration technique are widely used to improve the convergence rate of the iteration methods which can be solved effectively by constructing different iteration schemes.In this paper,some iteration methods for solving complex symmetric positive definite and complex symmetric indefinite linear systems are studied,and the convergence theory of the algorithm is analyzed.The main research work is as follows:For complex symmetric positive definite linear systems,we can transformed it into a equivalent block two-by-two real linear systems.Two different preconditioners including paracter a are introduced by using the preconditioned technique and then a two-step preconditioned block splitting iteration method is proposed(abbreviated as TPBS).It is proved theoretically that the TPBS iteration method is unconditionally convergent In order to make the TPBS iteration method can be applied to some special cases,we turn one parameter of the TPBS iteration method into two parameters and then construct a generalized TPBS iteration method(abbreviated as GTPBS).The convergence theory of the GTPBS iteration method is discussed,meanwhile,we propose a way to choose optimal iteration parameters α*and β*for the GTPBS iteration method.Finally,numerical experiments are given to testify the effectiveness and feasibility for the TPBS iteration method and GTPBS iteration method,and GTPBS iteration method performs better.For complex symmetric indefinite linear systems,by utilizing the SOR acceleration technique for the GPMHSS iteration method,we construct modified GPMHSS iteration method(abbreviated as MGPMHSS).The iteration scheme of MGPMHSS method is given and the corresponding converg-ence analysis is presented.According to the convergence condition,the selection method of relaxation factor δ is obtained.Numerical experiments illustrate that MGPMHSS method is feasible and effective for solving the complex symmetric indefinite linear systems by comparing the existing iteration methods.